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Module 1 · L11 of 15 ~40 min ⚡ +50 XP in Learn · +25 to complete

Dilations of Functions

Pinch to zoom on a photo and everything stretches or shrinks proportionally. In mathematics, dilations do exactly the same thing to graphs, stretching them away from an axis or compressing them toward it. Understanding dilations is the key to sketching almost every transformed function you will meet in the HSC.

Today's hook, Imagine the graph of $y = x^2$. If you change the equation to $y = 2x^2$, does the parabola become wider or narrower? What if you change it to $y = (2x)^2$? Can you describe in words what each change does to the shape of the graph?
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Orient and recall

Meet the destination, bring back what you already know, and gather the terms and formulas this lesson leans on.

Worksheets

Practise this lesson

Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question
01
Recall, your gut answer first
+5 XP warm-up

Imagine the graph of $y = x^2$. If you change the equation to $y = 2x^2$, do you think the parabola becomes wider or narrower? What if you change it to $y = (2x)^2$? Try to describe in words what each change does to the shape of the graph.

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Formula reference · this lesson
core notation
$y = af(x)$  , Vertical dilation by factor $a$ from the $x$-axis  ·  $a > 1$: stretch; $0 < a < 1$: compress
$y = f(bx)$  , Horizontal dilation by factor $\tfrac{1}{b}$ from the $y$-axis  ·  $b > 1$: compress; $0 < b < 1$: stretch
$y = f\!\left(\tfrac{x}{b}\right)$ Horizontal dilation by factor $b$ from the $y$-axis

Key insight: Vertical dilations affect $y$-coordinates directly. Horizontal dilations affect $x$-coordinates inversely: $f(bx)$ divides $x$-coordinates by $b$.

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What you'll master
Know

Key facts

  • $af(x)$ dilates vertically by factor $a$
  • $f(bx)$ dilates horizontally by factor $\frac{1}{b}$
  • How dilations affect coordinates of key points
Understand

Concepts

  • Why horizontal dilations are counter-intuitive (factor is $\frac{1}{b}$)
  • The difference between stretching and compressing
  • How dilations affect domain and range
Can do

Skills

  • Sketch dilated graphs from their equations
  • Write the equation of a dilated graph
  • Determine new coordinates after dilation
  • Combine dilations with translations and reflections
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Key terms
Dilation
Stretching or squashing a graph away from or towards an axis. Like this: $y = 3f(x)$ triples every height, so a point at height 2 moves to height 6.
Vertical dilation
Multiplying the whole function by a number, which changes heights only. Like this: $y = 2f(x)$ makes every peak twice as tall but no wider.
Horizontal dilation
Multiplying $x$ inside the function, which changes widths only. Like this: $y = f(2x)$ squashes the graph to half its width.
Dilation factor
The number the graph is scaled by. Like this: in $y = f(3x)$ the horizontal factor is $\tfrac{1}{3}$, because the graph becomes three times narrower.
Invariant point
A point that does not move under the transformation. Like this: under $y = af(x)$ every $x$-intercept stays put, because $a \times 0$ is still 0.
Stretch from an axis
The wording NESA uses for a dilation, naming the axis you move away from. Like this: a vertical dilation is a stretch from the $x$-axis.
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Establish it yourself with a graphing application

Work through the core explanation before applying it.

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Establish it yourself with a graphing application
+10 XP

The syllabus dot-point for this lesson begins "Establish using graphing applications", so this is the activity that earns it rather than being told the rule. Use Desmos, GeoGebra or any graphing calculator.

Task. Graph $y = \sqrt{x}$, then $y = 3\sqrt{x}$, then $y = \sqrt{3x}$, then $y = \sqrt{\tfrac{1}{3}x}$.

Predict before you plot. Before you look: predict which of these stretches away from an axis and which squashes towards one. Then check, and note which axis each dilation acts from.
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Vertical and horizontal dilations
core concept · +3 XP at end

A dilation stretches or compresses a graph by a scale factor from a fixed line (usually an axis). Unlike translations, which slide the graph without changing its shape, dilations actually change the distances between points, but they preserve the overall proportions of the graph.

Vertical Dilations: $y = af(x)$

When a constant is multiplied outside the function, every $y$-coordinate is multiplied by $a$. This stretches or compresses the graph vertically from the $x$-axis.

  • If $a > 1$, the graph is stretched away from the $x$-axis
  • If $0 < a < 1$, the graph is compressed toward the $x$-axis
  • If $a < 0$, there is also a reflection in the $x$-axis

For example, $y = 2x^2$ makes the parabola narrower because every $y$-value is doubled. The point $(1, 1)$ on $y = x^2$ moves to $(1, 2)$.

Horizontal Dilations: $y = f(bx)$

When a constant is multiplied inside the function, every $x$-coordinate is divided by $b$. This stretches or compresses the graph horizontally from the $y$-axis.

  • If $b > 1$, the graph is compressed toward the $y$-axis (dilation factor $\frac{1}{b}$)
  • If $0 < b < 1$, the graph is stretched away from the $y$-axis (dilation factor $\frac{1}{b} > 1$)
  • If $b < 0$, there is also a reflection in the $y$-axis

For example, $y = (2x)^2 = 4x^2$ compresses the parabola horizontally by factor $\frac{1}{2}$. The point $(1, 1)$ on $y = x^2$ moves to $(\frac{1}{2}, 1)$ because you need $x = \frac{1}{2}$ for $2x = 1$.

The horizontal dilation trap. Students often see $f(2x)$ and think "dilation by factor 2." It is not. The dilation factor is $\frac{1}{2}$. A larger number inside the brackets actually squeezes the graph closer to the $y$-axis. Think of it this way: to get the same output, you only need half the input, so the graph is squashed horizontally.

Alternative Form: $y = f\!\left(\frac{x}{b}\right)$

You will sometimes see dilations written as $f\!\left(\frac{x}{b}\right)$. In this form, the dilation factor is simply $b$ from the $y$-axis. This is often easier to read because the number in the denominator is the actual scale factor.

  • $y = f\!\left(\frac{x}{3}\right)$ is a horizontal dilation by factor 3
  • $y = f\!\left(\frac{x}{\frac{1}{2}}\right) = f(2x)$ is a horizontal dilation by factor $\frac{1}{2}$

Vertical dilation: $y = af(x)$, factor $a$ from the $x$-axis, $(x,y) \to (x, ay)$, $a > 1$: stretch; $0 < a < 1$: compress; Horizontal dilation: $y = f(bx)$, factor $\tfrac{1}{b}$ from the $y$-axis, $(x,y) \to \left(\tfrac{x}{b}, y\right)$

Pause, copy both dilation rules with their coordinate maps: $y = af(x)$ maps $(x,y) \to (x,ay)$; $y = f(bx)$ maps $(x,y) \to (x/b, y)$ into your book.

Did you get this? True or false: $y = f(3x)$ represents a horizontal dilation by factor 3 from the $y$-axis.

Quick check: The point $(4, 2)$ lies on $y = f(x)$. What is the corresponding point on $y = 2f(x)$?

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Effect on key features
core concept

We just saw that $y = af(x)$ scales $y$-values by $a$ and $y = f(bx)$ scales $x$-values by $\frac{1}{b}$. That raises a question: if every point shifts, how do specific features like intercepts, turning points, and asymptotes transform? This card answers it → vertical dilation leaves $x$-intercepts unchanged; horizontal dilation leaves the $y$-intercept unchanged but moves $x$-intercepts.

Dilations affect different features in specific ways:

  • Vertical dilation $af(x)$:
    • $x$-intercepts stay the same (where $y = 0$)
    • $y$-intercept is multiplied by $a$
    • Range is scaled by factor $a$
    • Domain is unchanged
  • Horizontal dilation $f(bx)$:
    • $y$-intercept stays the same (where $x = 0$)
    • $x$-intercepts are divided by $b$
    • Domain is scaled by factor $\frac{1}{b}$
    • Range is unchanged
y = 2 sin x stretches vertically (amplitude 2); y = sin 2x compresses horizontally (period π).

Vertical dilation: $x$-intercepts unchanged; $y$-intercept multiplied by $a$; range scaled by $a$; domain unchanged; Horizontal dilation: $y$-intercept unchanged; $x$-intercepts divided by $b$; domain scaled by $\tfrac{1}{b}$; range unchanged

Pause, copy the two-column feature table: vertical dilation ($x$-intercepts fixed, $y$-intercept $\times a$) vs horizontal dilation ($y$-intercept fixed, $x$-intercepts $\div b$) into your book.

Quick check: If $y = f(x)$ has a $y$-intercept at $(0, 3)$, what is the $y$-intercept of $y = f(5x)$?

Common mistakes · the 4 traps that cost marks

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Dodge the traps, then apply

Meet the mistakes that cost marks, then do it yourself.

1

Thinking $f(bx)$ dilates by factor $b$ horizontally

This is the single most common error with dilations. $f(2x)$ does not stretch by factor 2, it compresses by factor $\frac{1}{2}$. The number inside the bracket divides the $x$-coordinates, which makes the graph narrower, not wider.

✓ Fix: For $f(bx)$, always write the dilation factor as $\frac{1}{b}$. For $f(\frac{x}{b})$, the factor is $b$.

2

Confusing vertical and horizontal dilations

Students often describe $y = 2f(x)$ as a horizontal stretch and $y = f(2x)$ as a vertical stretch. The location of the coefficient determines which axis the dilation is from.

✓ Fix: Outside = vertical ($y$-direction). Inside = horizontal ($x$-direction).

3

Forgetting that dilations preserve the sign of intercepts

A vertical dilation by factor 2 will double a $y$-intercept, but it will not change its sign (unless the dilation factor is negative). Similarly, horizontal dilations do not change $y$-intercepts because $x = 0$ maps to $0$ regardless of the scale factor.

✓ Fix: $y$-intercepts are unaffected by horizontal dilations. $x$-intercepts are unaffected by vertical dilations.

4

Describing $f(2x)$ as "narrower" without specifying the axis

In exam questions, vague descriptions like "the graph is narrower" may not score full marks. You must specify whether the dilation is from the $x$-axis or the $y$-axis.

✓ Fix: Always say "vertical dilation by factor ... from the $x$-axis" or "horizontal dilation by factor ... from the $y$-axis."

Worked examples · reveal as you go

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Work examples end to end

Follow the reasoning through complete worked solutions.

Worked example 1 · describing a dilation +5 XP on full reveal

Describe the transformation that maps $y = f(x)$ to $y = 3f(x)$.

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Identify where the coefficient is
The 3 is outside the function, multiplying the output.
2
Determine the type and factor
Outside = vertical dilation. The factor is the coefficient itself: 3.
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Vertical dilation by factor 3 from the $x$-axis ✓
Since $3 > 1$, the graph is stretched away from the $x$-axis.
Step 1 of 3
Worked example 2 · horizontal dilation +5 XP on full reveal

Describe the transformation that maps $y = f(x)$ to $y = f(2x)$.

1
Identify where the coefficient is
The 2 is inside the function, multiplying the input $x$.
2
$$\text{Dilation factor} = \frac{1}{2}$$
Inside = horizontal dilation, and the factor is the reciprocal of the coefficient.
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Horizontal dilation by factor $\frac{1}{2}$ from the $y$-axis ✓
Since $\frac{1}{2} < 1$, the graph is compressed toward the $y$-axis.
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Worked example 3 · finding dilated coordinates +5 XP on full reveal

The graph of $y = f(x)$ passes through $(2, 3)$ and $(6, -1)$. Find the corresponding points on $y = 2f\!\left(\frac{x}{3}\right)$.

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Identify both transformations
$2$ outside = vertical dilation by factor 2. $\frac{x}{3}$ inside = horizontal dilation by factor 3.
2
Apply the horizontal dilation: multiply $x$-coordinates by 3.
$(2, 3) \to (6, 3)$ and $(6, -1) \to (18, -1)$
3
Apply the vertical dilation: multiply $y$-coordinates by 2.
$(6, 3) \to (6, 6)$ and $(18, -1) \to (18, -2)$
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$(6, 6)$ and $(18, -2)$ ✓
Apply horizontal dilation first (multiply $x$ by 3), then vertical (multiply $y$ by 2).
Step 1 of 4

Fill the blanks: drag each token into the matching blank.

outside inside 1/b b

For $y = af(x)$, the coefficient $a$ is ___ the function, vertical dilation. For $y = f(bx)$, the coefficient $b$ is ___ horizontal dilation by factor ___. For $y = f(\frac{x}{b})$, the horizontal dilation factor is ___.

Activity 1, Describe the dilation

For each equation, describe the dilation from $y = f(x)$. State whether it is vertical or horizontal, and give the dilation factor.

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Dodge the traps, then apply

Meet the mistakes that cost marks, then do it yourself.

1

$y = 4f(x)$

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$y = f(3x)$

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3

$y = f\!\left(\frac{x}{2}\right)$

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$y = \frac{1}{3}f(x)$

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Odd one out: Which of these is a horizontal dilation?

Quick-fire practice · 5 reps +2 XP per reveal

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Drill it, then lock it in

Run the quick drill and copy the summary into your book.

1

Describe the dilation that maps $y = f(x)$ to $y = \frac{1}{2}f(x)$.

2

If $(6, 4)$ lies on $y = f(x)$, find the corresponding point on $y = f(3x)$.

3

The $y$-intercept of $y = f(x)$ is $(0, -2)$. What is the $y$-intercept of $y = 5f(x)$?

4

$y = f(x)$ has an $x$-intercept at $(4, 0)$. What is the $x$-intercept of $y = f\!\left(\frac{x}{2}\right)$?

5

Explain why $y = f(2x)$ represents a horizontal compression by factor $\frac{1}{2}$, not a stretch by factor 2.

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Revisit your thinking

Earlier you were asked: If you change $y = x^2$ to $y = 2x^2$, does the parabola become wider or narrower? What about $y = (2x)^2$?

$y = 2x^2$ is a vertical dilation by factor 2. It stretches the graph away from the $x$-axis, making the parabola appear narrower because the $y$-values grow faster. $y = (2x)^2 = 4x^2$ is a horizontal dilation by factor $\frac{1}{2}$. It compresses the graph toward the $y$-axis, which also makes the parabola appear narrower. Both transformations change the shape, but they do so in different directions, one vertically, one horizontally. For parabolas, these effects can look similar, but for more complex functions the difference between vertical and horizontal dilations is dramatic.

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Multiple choice

Answer the drill bank and rate your confidence.

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Multiple choice
+5 XP per correct · +25 XP all-correct

Pick your answer, then rate your confidence, that tells the system what to drill next.

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Short answer

Write full responses, then check them against the model answers.

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Short answer
UnderstandBand 32 marks

Q8. Explain why $y = f(2x)$ represents a horizontal compression by factor $\frac{1}{2}$, not a stretch by factor 2. Use the idea of inputs and outputs in your explanation.

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ApplyBand 44 marks

Q9. The graph of $y = f(x)$ passes through $(1, 2)$, $(3, 5)$, and $(6, -1)$. Find the corresponding points on: (a) $y = 2f(x)$ (b) $y = f(3x)$ (c) $y = 2f(3x)$

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AnalyseBand 54 marks

Q10. Consider $f(x) = x^2$. (a) Write the equation of the graph after a vertical dilation by factor 2 from the $x$-axis. (b) Write the equation of the graph after a horizontal dilation by factor $\frac{1}{2}$ from the $y$-axis. (c) Show algebraically that these two transformations produce the same equation for this particular function. Explain why this does not happen for all functions.

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📖 Comprehensive answers (click to reveal)

Multiple choice, drill bank

MC answers and feedback are shown inline as you complete each question. Use the retry button to attempt a fresh set.

Vertical dilation. $3f(x)$ is a vertical dilation by factor 3.

Horizontal dilation factor. $f(2x)$ is a horizontal dilation by factor $\frac{1}{2}$.

Dilating a single point. $(2, 3) \to (2, 6)$.

Horizontal dilation on a point. $f(\frac{x}{3})$ dilates horizontally by factor 3: $(6, 1) \to (18, 1)$.

Reading $f(x/2)$. $f(\frac{x}{2})$ means horizontal dilation by factor 2.

Activity 1, Describe the dilation model answers

1. Vertical dilation by factor 4 from the $x$-axis

2. Horizontal dilation by factor $\frac{1}{3}$ from the $y$-axis

3. Horizontal dilation by factor 2 from the $y$-axis

4. Vertical dilation by factor $\frac{1}{3}$ from the $x$-axis

Short answer model answers

Q8 (2 marks): For $f(2x)$ to produce the same output as $f(1)$, the input must be $x = \frac{1}{2}$ [1]. This means every point moves halfway toward the $y$-axis, so the graph is compressed by factor $\frac{1}{2}$, not stretched [1].

Q9 (4 marks):

(a) $(1, 4)$, $(3, 10)$, $(6, -2)$ [1–2 marks]
(b) $(\frac{1}{3}, 2)$, $(1, 5)$, $(2, -1)$ [1–2 marks]
(c) $(\frac{1}{3}, 4)$, $(1, 10)$, $(2, -2)$ [1 mark]

Q10 (4 marks):

(a) $y = 2x^2$ [1]
(b) Horizontal dilation by factor $\frac{1}{2}$ means $f(2x) = (2x)^2 = 4x^2$, wait, this is not the same as $2x^2$.
The expected answer: vertical dilation by 2 gives $2x^2$; horizontal compression by $\frac{1}{2}$ gives $(2x)^2 = 4x^2$. These are different [1]. However, if the student uses $f(\frac{x}{\sqrt{2}})$ then $(\frac{x}{\sqrt{2}})^2 = \frac{x^2}{2} \cdot 2 = x^2$... For full marks, explain that for $f(x) = x^2$, the relationship $af(x) = f(\sqrt{a}x)$ holds because of the even exponent [1–2]. For a general function this equivalence does not hold [1].

Module overview · Maths Advanced